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Mirrors > Home > MPE Home > Th. List > divcan4i | Structured version Visualization version GIF version |
Description: A cancellation law for division. (Contributed by NM, 18-May-1999.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
divclz.2 | ⊢ 𝐵 ∈ ℂ |
divcl.3 | ⊢ 𝐵 ≠ 0 |
Ref | Expression |
---|---|
divcan4i | ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcl.3 | . 2 ⊢ 𝐵 ≠ 0 | |
2 | divclz.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | divclz.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | 2, 3 | divcan4zi 11181 | . 2 ⊢ (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 ≠ wne 2967 (class class class)co 6976 ℂcc 10333 0cc0 10335 · cmul 10340 / cdiv 11098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 |
This theorem is referenced by: mvllmuli 11274 4bc2eq6 13504 bpoly3 15272 ang180lem1 25088 bposlem8 25569 2lgslem3d 25677 2lgsoddprmlem3c 25690 2lgsoddprmlem3d 25691 decdiv10 30325 dpmul4 30343 areaquad 39225 6even 43250 8even 43252 mvlrmuli 44251 |
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